Finding Missing Values Using Equations A Step-by-Step Guide

Hey guys! Let's dive into this math problem together. We've got a table with some X and Y values, and we need to figure out the missing piece of the puzzle. Are you ready to put on your math hats and get started?

The Problem

We have a table that looks like this:

The question asks us to find the missing value of Y when X is 3. We're given the equation Y = 4 - 3x + x³ to help us solve this. So, let's break it down step by step and find that missing number!

Breaking Down the Equation

Before we jump into plugging in the value of X, let's take a closer look at the equation: Y = 4 - 3x + x³. This equation tells us how Y changes based on the value of X. It's a cubic equation because of the x³ term, which means it can have a slightly curvy graph. Don't let that scare you, though! We're just dealing with one specific point here.

The first part of the equation is the constant 4. This means that no matter what X is, we always start with 4. The second part is -3x, which means we multiply X by -3. This part will change Y depending on whether X is positive or negative. And finally, the x³ part means we take X and multiply it by itself three times (X * X * X). This part can have a big impact on Y, especially when X gets larger.

To find the missing value, we're going to substitute X = 3 into this equation. This will tell us exactly what Y is when X is 3. It's like we're following a recipe – just plug in the right ingredients (the value of X) and follow the instructions (the equation) to get our final result (the value of Y).

Step-by-Step Solution

Okay, let's get our hands dirty and solve this thing! We know that X = 3, and we want to find Y using the equation Y = 4 - 3x + x³.

1. Substitute X with 3: Y = 4 - 3(3) + (3)³

   We've replaced every 'x' in the equation with the number 3. Now we just need to do the math.

2. Calculate the exponent (3³): Y = 4 - 3(3) + 27

   3³ means 3 * 3 * 3, which equals 27. We've simplified the exponent part.

3. Perform the multiplication (-3 * 3): Y = 4 - 9 + 27

   -3 multiplied by 3 is -9. Now we have a simpler equation with just addition and subtraction.

4. Add and subtract from left to right: Y = 4 - 9 + 27 Y = -5 + 27 Y = 22

   First, we did 4 - 9, which gave us -5. Then, we added 27 to -5, which gave us the final answer: 22!

The Answer

So, the missing value in the table is 22. When X is 3, Y is 22. Awesome job, guys! We've cracked the code and found the missing piece.

Why This Matters

You might be thinking, "Okay, we found a number, but why does this matter?" Well, these kinds of equations and tables show up in all sorts of real-world situations. They can help us understand relationships between things, like how the speed of a car affects the distance it travels, or how the amount of fertilizer affects the growth of a plant. Understanding how to work with these equations is a valuable skill.

In this case, we used the equation to predict the value of Y for a specific value of X. This is a fundamental concept in many fields, including science, engineering, and economics. Being able to plug in values and solve for unknowns is a crucial part of problem-solving.

Practice Makes Perfect

The best way to get comfortable with these kinds of problems is to practice! Try changing the equation or the X values and see what happens. You can even create your own tables and try to find the missing values. The more you practice, the easier it will become. You'll start to see patterns and understand how the different parts of the equation work together.

Maybe try graphing the equation Y = 4 - 3x + x³ to see what it looks like. This can give you a visual understanding of how Y changes as X changes. There are lots of online tools that can help you graph equations, so give it a try!

Common Mistakes to Avoid

When working with equations like this, there are a few common mistakes that people make. Knowing about these pitfalls can help you avoid them.

• Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS). This means doing parentheses/brackets first, then exponents/orders, then multiplication and division, and finally addition and subtraction. If you don't follow the order of operations, you might get the wrong answer.
• Sign Errors: Pay close attention to negative signs. A simple sign error can throw off your entire calculation. Double-check your work, especially when you're dealing with subtraction or negative numbers.
• Substituting Correctly: Make sure you're substituting the value of X into the equation correctly. It's easy to accidentally substitute into the wrong place or forget a term. Take your time and double-check that you've replaced every 'x' with the correct value.
• Calculation Errors: Even a small arithmetic error can lead to the wrong answer. Use a calculator if you need to, and double-check your calculations to make sure they're accurate.

By being aware of these common mistakes, you can increase your chances of getting the right answer and build your confidence in solving these types of problems.

Let's Recap

Okay, let's do a quick recap of what we've learned. We started with a table and an equation, and our mission was to find the missing value. We broke down the equation, substituted the value of X, and followed the order of operations to calculate Y. We found that when X is 3, Y is 22. We also talked about why this kind of problem-solving is important and how it can be applied in the real world.

Remember, math is like a puzzle, and each piece fits together in a specific way. By taking things step by step and practicing, you can become a math master in no time! Keep up the great work, guys!